Question 284·Hard·Math

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In the figure, all three segments with matching tick marks have equal lengths. Which choice gives the value of $x$ ?

Matching tick marks reveal isosceles triangles, which have two equal sides. In each triangle, match the equal sides to the angles opposite them. Then use the straight line through the triangles and the given whole angle to build an equation; a Desmos regression can solve it. Watch for the trap at the end: the angle outside the triangle is not the small angle beside it.

Hints

Hint 1
Start in triangle $BDC$ . Since $BD=DC$ , the base angles opposite those sides are equal at $B$ and $C$ . Call each $t^\circ$ . What angle do those two angles give you outside the triangle at $D$ ?
Hint 2
Because $A$ , $D$ , and $C$ lie on a line, $\angle ADB$ is an exterior angle of triangle $BDC$ . It equals the two far interior angles added together. Which angle in triangle $ABD$ matches it?
Hint 3
The given angle at $B$ has two pieces, one on each side of $BD$ . Add them to make an equation, use a Desmos regression to find the small angle, and then return to the exterior angle at $C$ .

Step-by-step

Follow the equal sides through both triangles
Step 1
Match the angles in triangle BDC
The matching ticks give $BD=DC$ . Equal sides face equal angles: $BD$ faces $\angle BCD$ , and $DC$ faces $\angle DBC$ . Call each base angle $t^\circ$ , so they have the same measure:
$m\angle DBC=m\angle BCD=t^\circ.$
Step 2
Find the angle at D in the other triangle
Since $A$ , $D$ , and $C$ lie on a straight line, $\angle ADB$ is an exterior angle of triangle $BDC$ . An exterior angle equals the two far interior angles added together:
$m\angle ADB=(t+t)^\circ=2t^\circ.$
Step 3
Use the second pair of equal sides
The ticks also give $AB=BD$ . In triangle $ABD$ , $AB$ faces $\angle ADB$ and $BD$ faces $\angle BAD$ , so those opposite angles match:
$m\angle BAD=m\angle ADB=2t^\circ.$
Step 4
Find the remaining angle in triangle ABD
A triangle’s angles add to $180^\circ$ . Subtract the two $2t^\circ$ angles to find the angle at $B$ :
$m\angle ABD=180^\circ-2t^\circ-2t^\circ.$
Combine the matching terms:
$m\angle ABD=(180-4t)^\circ.$
Step 5
Make an equation from the whole angle at B
Segment $BD$ splits the given $105^\circ$ angle into $\angle ABD$ and $\angle DBC$ . Add both pieces, not only the angle in triangle $ABD$ :
$(180-4t)+t=105.$
Step 6
Solve for the small angle
Combine the $t$ terms:
$180-3t=105.$
Type $180-3t_1\sim105$ in Desmos. The subscript makes $t_1$ the unknown, and $\sim$ asks Desmos to fit the two sides. Under PARAMETERS, it shows $t_1=25$ . So $\angle BCD$ , one of the base angles, measures $25^\circ$ .
Step 7
Find the exterior angle at C
Ray $CE$ continues the straight line through $C$ , so $\angle BCD$ and the exterior angle $x^\circ$ add to $180^\circ$ . Type $180-t_1$ below the regression; Desmos prints $155$ . The $25^\circ$ angle is inside the triangle, but $x=155$ is the angle asked for. Choice D.

Question 284·Hard·Math

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